For x is greater than three, this is going to be the exact same thing as x minus three over x minus three because if x is greater than three, the numerator's going to be positive, you take the absolute value of that, you're not going to change its value so you get this right over here or, if we were to re-write it.if we were to re-write it, this is equal to, for x is greater than three, you're going to have f(x) is equal to one for x is greater than three Similarly, let's think about what happens when x is less than three When x is less than three, well, x minus three is going to be a negative number When you take the absolute value of that, you're essentially negating it so it's going to be the negative of x minus three over x minus three or if you were to simplify these two things, for any value as long as x doesn't equal three this part right over her simplifies to one, so you are left with a negative one negative one for x is less than three I encourage you, if you don't believe what I just said, try it out with some numbers Try out some numbers: 3.1, 3.001, 3.5, 4, 7 Any number greater than three, you're going to get one You're going to get the same thing divided by the same thing and try values for x less than three: you're going to get negative one no matter what you try So let's visualise this function now So, now you draw some axes. Well, if I take the absolute value it's going to be the exact same thing, so let me. I'm going to get a positive value up here and then I'm. When x is greater than three, what does this function simplify to? Well, whatever I get up here, I'm just taking the.
Let's say that f(x) is equal to the absolute value of x minus three over x minus three and what I'm curious about is the limit of f(x) as x approaches three and just from an inspection you can see that the function is not defined when x is equal to three - you get zero over zero: it's not defined So to answer this question let's try to re-write the same exact function definition slightly differently So let's say f(x) is going to be equal to - and I'm going to think of two cases: I'm going to think of the case when x is greater than three and when x is less than three So when x is - I'll do this in two different colors actually When x - I'll do it in green - that's not green When x is greater than three.
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